CHAPTER 14
MANAGING BOND PORTFOLIOS
1. The percentage bond price change will be:
– y = – .005 = –.0327 or a 3.27% decline.
2. Computation of duration:
a.YTM = 6%
(1) / (2) / (3) / (4) / (5)Time until payment
(years) / Payment / Payment discounted at 6% / Weight of each payment / Column (1) Column (4)
1 / 60 / 56.60 / .0566 / .0566
2 / 60 / 53.40 / .0534 / .1068
3 / 1060 / 890.00 / .8900 / 2.6700
Column Sum / 1000.00 / 1.0000 / 2.8334
Duration = 2.833 years
b.YTM = 10%
(1) / (2) / (3) / (4) / (5)Time until payment
(years) / Payment / Payment discounted at 10% / Weight of each payment / Column (1) Column (4)
1 / 60 / 54.55 / .0606 / .0606
2 / 60 / 49.59 / .0551 / .1102
3 / 1060 / 796.39 / .8844 / 2.6532
Column Sum / 900.53 / 1.0000 / 2.8240
Duration = 2.824 years, which is less than the duration at the YTM of 6%.
3.For a semiannual 6% coupon bond selling at par, we use parameters c = 3% per half-year period, y = 3%, T = 6 semiannual periods. Using Rule 8, we find that:
D = (1.03/.03) [ 1 – (1/1.03)6]
= 5.58 half year periods
= 2.79 years
If the bond’s yield is 10%, use Rule 7, setting the semiannual yield to 5%, and semiannual coupon to 3%:
D = –
= 21 – 15.448 = 5.552 half-year periods = 2.776 years
4. a.Bond B has a higher yield to maturity than bond A since its coupon payments and maturity are equal to those of A, while its price is lower. (Perhaps the yield is higher because of differences in credit risk.) Therefore, its duration must be shorter.
b. Bond A has a lower yield and a lower coupon, both of which cause it to have a longer duration than B. Moreover, A cannot be called, which makes its maturity at least as long as that of B, which generally increases duration.
5.t CFPV(CF)Weightw t
110 9.09 .786 .786
5 4 2.48 .2141.070
11.571.0001.856
a.D = 1.856 years = required maturity of zero coupon bond
b.The market value of the zero must be $11.57 million, the same as the market value of the obligations. Therefore, the face value must be $11.57 (1.10)1.856 = $13.81 million.
6.a.The call feature provides a valuable option to the issuer, since it can buy back the bond at a given call price even if the present value of the scheduled remaining payments is more than the call price. The investor will demand, and the issuer will be willing to pay, a higher yield on the issue as compensation for this feature.
b.The call feature will reduce both the duration (interest rate sensitivity) and the convexity of the bond. The bond will not experience as large a price increase if interest rates fall. Moreover the usual curvature that would characterize a straight bond will be reduced by a call feature. The price-yield curve (see Figure 13.7) flattens out as the interest rate falls and the option to call the bond becomes more attractive. In fact, at very low interest rates, the bond exhibits “negative convexity.”
7.Choose the longer-duration bond to benefit from a rate decrease.
a.The Aaa-rated bond will have the lower yield to maturity and the longer duration.
b.The lower-coupon bond will have the longer duration and more de facto call protection.
c.Choose the lower coupon bond for its longer duration.
8.a. The price decreases by 10 .01 $800 = $80.00
b. ½ 120 (.015)2 = .0135 = 1.35%
c. 9/1.10 = 8.18
d.(i)
e.(i)
f.(iii)
9.You should buy the 3-year bond because it will offer a 9% holding-period return over the next year, which is greater than the return on either of the other bonds.
Maturity:1 year2 years 3 years
YTM at beginning of year 7% 8% 9%
Beginning of year prices$1009.35$1000.00$974.69
Prices at year end (at 9% YTM)$1000.00$ 990.83$982.41
Capital gain –$ 9.35 –$ 9.17$ 7.72
Coupon$ 80.00$ 80.00$ 80.00
1-year total $ return$ 70.65$ 70.83$ 87.72
1-year total rate of return 7.00% 7.08% 9.00%
The 3-year bond provides the greatest holding period return.
10.a.Modified duration =
If the Macaulay duration is 10 years and the yield to maturity is 8%, then the modified duration equals 10/1.08 = 9.26 years.
b.For option-free coupon bonds, modified duration is better than maturity as a measure of the bond’s sensitivity to changes in interest rates. Maturity considers only the final cash flow, while modified duration includes other factors such as the size and timing of coupon payments and the level of interest rates (yield to maturity). Modified duration, unlike maturity, tells us the approximate percentage change in the bond price for a given change in yield to maturity.
c.i.Modified duration increases as the coupon decreases.
ii.Modified duration decreases as maturity decreases.
d.Convexity measures the curvature of the bond’s price-yield curve. Such curvature means that the duration rule for bond price change (which is based only on the slope of the curve at the original yield) is only an approximation. Adding a term to account for the convexity of the bond will increase the accuracy of the approximation. That convexity adjustment is the last term in the following equation:
= –D*y + Convexity (y)2
11.a. PV of the obligation = $10,000 Annuity factor (8%, 2) = $17,832.65
Duration = 1.4808 years, which can be verified from rule 6 or a table like Table 13.3.
b.To immunize my obligation I need a zero-coupon bond maturing in 1.4808 years. Since the present value must be $17,832.65, the face value (i.e., the future redemption value) must be 17,832.65 1.081.4808 or $19,985.26.
c.If the interest rate increases to 9%, the zero-coupon bond would fall in value to
= $17,590.92
and the present value of the tuition obligation would fall to $17,591.11. The net position decreases in value by $.19.
If the interest rate falls to 7%, the zero-coupon bond would rise in value to
= $18,079.99
and the present value of the tuition obligation would rise to $18,080.18. The net position decreases in value by $.19.
The reason the net position changes at all is that, as the interest rate changes, so does the duration of the stream of tuition payments.
12.a.(i) Current yield = Coupon/Price = $70/$960 = 0.0729 = 7.29%
(ii) YTM = 3.993% semiannually or 7.986% annual bond equivalent yield.
On a financial calculator, enter: n = 10; PV = –960; FV = 1000; PMT = 35
Compute the interest rate.
(iii) Horizon yield or realized compound yield is 4.166% (semiannually), or 8.332% annual bond equivalent yield. To obtain this value, first find the future value (FV) of reinvested coupons and principal. There will be six payments of $35 each, reinvested semiannually at 3% per period. On a financial calculator, enter:
PV = 0; PMT = $35; n = 6; i = 3%. Compute: FV = $226.39
Three years from now, the bond will be selling at the par value of $1,000 because the yield to maturity is forecast to equal the coupon rate. Therefore, total proceeds in three years will be $1,226.39.
Then find the rate (yrealized) that makes the FV of the purchase price equal to $1,226.39:
$960 (1 + yrealized)6 = $1,226.39 yrealized = 4.166% (semiannual)
b.Shortcomings of each measure:
(i) Current yield does not account for capital gains or losses on bonds bought at prices other than par value. It also does not account for reinvestment income on coupon payments.
(ii) Yield to maturity assumes the bond is held until maturity and that all coupon income can be reinvested at a rate equal to the yield to maturity.
(iii) Horizon yield or realized compound yield is affected by the forecast of reinvestment rates, holding period, and yield of the bond at the end of the investor's holding period.
Note: This criticism of horizon yield is a bit unfair: while YTM can be calculated without explicit assumptions regarding future YTM and reinvestment rates, you implicitly assume that these values equal the current YTM if you use YTM as a measure of expected return.
13.a.i. The effective duration of the 4.75% Treasury security is:
ii. The duration of the portfolio is the weighted average of thedurations of the individual bonds in the portfolio:
Portfolio Duration = w1D1 + w2D2 + w3D3 + … + wkDk
where
wi= market value of bond i/market value of the portfolio
Di = duration of bond i
k = number of bonds in the portfolio
The effective duration of the bond portfolio is calculated as follows:
[($48,667,680/$98,667,680) × 2.15] + [($50,000,000/$98,667,680) × 15.26] = 8.79
b.VanHusen’s remarks wouldbe correct if there were a small, parallel shift in yields. Duration is a first (linear) approximation only for small changes in yield. Forlarger changes in yield, the convexity measure is needed in order to approximate the change inprice that is not explained by duration. Additionally, portfolio duration assumes that allyields change by the same number of basis points (parallel shift), so any non-parallel shiftin yields would result in a difference in the price sensitivity of the portfolio compared to the price sensitivity of asingle security having the same duration.
14.a.PV of obligation = $2 million/.16 = $12.5 million.
Duration of obligation = 1.16/.16 = 7.25 years
Call w the weight on the 5-year maturity bond (which has duration of 4 years). Then
w 4 + (1 – w) 11 = 7.25
which implies that w = .5357.
Therefore, .5357 $12.5 = $6.7 million in the 5-year bond and
.4643 $12.5 = $5.8 million in the 20-year bond.
b.The price of the 20-year bond is:
60 Annuity factor(16%, 20) + 1000 PV factor(16%, 20) = $407.12.
Therefore, the bond sells for .4071 times its par value, and
Market value = Par value .4071
$5.8 million = Par value .4071
Par value = $14.25 million
Another way to see this is to note that each bond with par value $1000 sells for $407.11. If total market value is $5.8 million, then you need to buy approximately 14,250 bonds, resulting in total par value of $14,250,000.
15.a.The duration of the perpetuity is 1.05/.05 = 21 years. Let w be the weight of the zero-coupon bond. Then we find w by solving:
w 5 + (1 – w) 21 = 10
21 – 16w = 10 w = 11/16 = .6875
Therefore, your portfolio would be 11/16 invested in the zero and 5/16 in the perpetuity.
b. The zero-coupon bond now will have a duration of 4 years while the perpetuity will still have a 21-year duration. To get a portfolio duration of 9 years, which is now the duration of the obligation, we again solve for w:
w 4 + (1 – w) 21 = 9
21 – 17w = 9
w = 12/17 or .7059
So the proportion invested in the zero increases to 12/17 and the proportion in the perpetuity falls to 5/17.
16a.From Rule 6, the duration of the annuity if it were to start in 1 year would be
– = 4.7255 years
Because the payment stream starts in 5 years, instead of one year, we must add 4 years to the duration, resulting in duration of 8.7255 years.
b.The present value of the deferred annuity is
= $41,968.
Call w the weight of the portfolio in the 5-year zero. Then
5w + 20(1 – w) = 8.7255
which implies that w = .7516 so that the investment in the 5-year zero equals
.7516 $41,968 = $31,543.
The investment in 20-year zeros is .2484 $41,968 = $10,425.
These are the present or market values of each investment. The face value of each is the future value of the investment.
The face value of the 5-year zeros is
$31,543 (1.10)5 = $50,800
meaning that between 50 and 51 zero coupon bonds, each of par value $1,000, would be purchased. Similarly, the face value of the 20-year zeros would be:
$10,425 (1.10)20 = $70,134.
17.a.The Aa bond initially has a higher YTM (yield spread of 40 b.p. versus 31 b.p.), but it is expected to have a widening spread relative to Treasuries. This will reduce the rate of return. The Aaa spread is expected to be stable. Calculate comparative returns as follows:
Incremental return over Treasuries =
Incremental yield spread (Change in spread duration)
Aaa bond:31 bp (0 3.1 years) = 31 bp
Aa bond:40 bp (10 bp 3.1 years) = 9 bp
Therefore, choose the Aaa bond.
b.Other variables to be considered:
Potential changes in issue-specific credit quality. If the credit quality of the bonds changes, spreads relative to Treasuries will also change.
Changes in relative yield spreads for a given bond rating. If quality spreads in the general bond market change because of changes in required risk premiums, the yield spreads of the bonds will change even if there is no change in the assessment of the credit quality of these particular bonds.
Maturity effect. As bonds near their maturity, the effect of credit quality on spreads can also change. This can affect bonds of different initial credit quality differently.
18.Using a financial calculator, the actual price of the bond as a function of yield to maturity is:
Yield to maturityPrice
7%$1620.45
8%$1450.31
9% $1308.21
Using the Duration Rule, assuming yield to maturity falls to 7%
Predicted price change = – y P0
= – (.01) 1450.31 = $154.97
Therefore, predicted new price = 154.97 + 1450.31 = $1605.28
The actual price at a 7% yield to maturity is $1620.45. Therefore,
% error = =.0094 = .94 % (approximation is too low)
Using the Duration Rule, assuming yield to maturity increases to 9%
Predicted price change = – y P0
= – .01 1450.31 = –$154.97
Therefore, predicted new price = –154.97 + 1450.31 = $1295.34
The actual price at a 9% yield to maturity is $1308.21. Therefore,
% error = = .0098 = .98 % (approximation is too low)
Using Duration-with-Convexity Rule, assuming yield to maturity falls to 7%
Predicted price change = [(– y) + (0.5 Convexity y2)] P0
=– (.01) + 0.5 192.4 (.01)2] 1450.31 = $168.92
Therefore, predicted price = 168.92 + 1450.31 = $1619.23
The actual price at a 7% yield to maturity is $1620.45. Therefore,
% error = = .00075 = .075% (approximation is too low)
Using Duration-with-Convexity Rule, assuming yield to maturity rises to 9%
Predicted price change = [(– y) + (0.5 Convexity y2)] P0
= – .01 + 0.5 192.4 (0.01)2] 1450.31 = –$141.02
Therefore, predicted price = –141.02 + 1450.31 = $1309.29
The true price at a 9% yield to maturity is $1308.21. Therefore,
% error = .00083 = .083% (approximation is too high)
Conclusion: the duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error using only duration to estimate the price change.
19.a.The price of the zero coupon bond ($1000 face value) selling at a yield to maturity of 8% is $374.84 and that of the coupon bond is $774.84.
At a YTM of 9% the actual price of the zero coupon bond is $333.28 and that of the coupon bond is $691.79.
Zero coupon bond
Actual % loss = = –.1109, an 11.09% loss
The percentage loss predicted by the duration-with-convexity rule is:
Predicted % loss= [( –11.81) .01 + 0.5 150.3 (0.01)2]
= –.1106, an 11.06% loss
Coupon bond
Actual % loss = = –.1072, a 10.72% loss
The percentage loss predicted by the duration-with-convexity rule is:
Predicted % loss= [( –11.79) .01 + 0.5 231.2 (0.01)2]
= –.1063, a 10.63% loss
b.Now assume yield to maturity falls to 7%. The price of the zero increases to $422.04, and the price of the coupon bond increases to $875.91.
Zero coupon bond
Actual % gain = = .1259, a 12.59% gain
The percentage gain predicted by the duration-with-convexity rule is:
Predicted % gain = [( –11.81) (–.01) + 0.5 150.3 (0.01)2 ]
= .1256, an 12.56% gain
Coupon bond
Actual % gain = = .1304, a 13.04% gain
The percentage gain predicted by the duration-with-convexity rule is:
Predicted % gain = [ (–11.79) (–.01) + 0.5 231.2 (0.01)2]
= .1295, a 12.95% gain
c.The 6% coupon bond—which has higher convexity—outperforms the zero regardless of whether rates rise or fall. This can be seen to be a general property using the duration-with-convexity formula: the duration effects on the two bonds due to any change in rates will be equal (since their durations are virtually equal), but the convexity effect, which is always positive, will always favor the higher convexity bond. Thus, if the yields on the bonds always change by equal amounts, as we have assumed in this example, the higher convexity bond will always outperform a lower convexity bond with equal duration and initial yield to maturity.
d.This situation cannot persist. No one would be willing to buy the lower convexity bond if it always underperforms the other bond. Its price will fall and its yield to maturity will rise. Thus, the lower convexity bond will sell at a higher initial yield to maturity. That higher yield is compensation for lower convexity. If rates change by only a little, the higher yield-lower convexity bond will do better; if rates change by a lot, the lower yield-higher convexity bond will do better.
20.a.The following spreadsheet shows that the convexity of the bond is 64.933. The present value of each cash flow is obtained by discounting at 7%. (since the bond has a 7% coupon and sells at par, its YTM must be 7%.) Convexity equals the sum of the last column, 7434.175, divided by [P (1 + y)2] = 100 (1.07)2.
Time (t) / Cash flow, CF / PV(CF) / t + t2 / (t + t2) x PV(CF)1 / 7 / 6.542 / 2 / 13.084
2 / 7 / 6.114 / 6 / 36.684
3 / 7 / 5.714 / 12 / 68.569
4 / 7 / 5.340 / 20 / 106.805
5 / 7 / 4.991 / 30 / 149.727
6 / 7 / 4.664 / 42 / 195.905
7 / 7 / 4.359 / 56 / 244.118
8 / 7 / 4.074 / 72 / 293.333
9 / 7 / 3.808 / 90 / 342.678
10 / 107 / 54.393 / 110 / 5983.271
Sum: / 100.000 / 7434.175
Convexity: / 64.933
The duration of the bond is (from rule 8):
D = [1 – ] = 7.515 years
b.If the yield to maturity increases to 8%, the bond price will fall to 93.29% of par value, a percentage decline of 6.71%.
c.The duration rule would predict a percentage price change of
– .01 = – .01 = – .0702 = – 7.02%
This overstates the actual percentage decline in price by .31%.
d.The duration with convexity rule would predict a percentage price change of
– .01 + .5 64.933 (.01)2 = .0670 = –6.70%
which results in an approximation error of only .01%, far smaller than the error using the duration rule.
21.a.% price change = Effective duration Change in YTM (%)
CIC: 7.35 (.50%) = 3.675%
PTR:5.40 (.50%) = 2.700%
b.There is no reinvestment income, since we are asked to calculate horizon return over a period of only one coupon period.
Horizon return =
CIC: = .06806 = 6.806%
PTR: = .05971 = 5.971%
c.Notice that CIC is non-callable but PTR is callable. Therefore, CIC will have positive convexity, while PTR will have negative convexity. Thus, the convexity correction to the duration approximation will be positive for CIC and negative for PTR.
22.The economic climate is one of impending interest rate increases. Hence, we will want to shorten portfolio duration.
a. Choose the short maturity (2012) bond.
b. The Arizona bond likely has lower duration. The Arizona coupons are slightly lower, but the Arizona yield is substantially higher.
c.Choose the 12 3/8 coupon bond. The maturities are about equal, but the 12 3/8 coupon is much higher, resulting in a lower duration.
d. The duration of the Shell bond will be lower if the effect of the higher yield to maturity and earlier start of sinking fund redemption dominates its slightly lower coupon rate.
e. The floating rate bond has a duration that approximates the adjustment period, which is only 6 months.
23.a.A manager who believes that the level of interest rates will change should engage in a rate anticipation swap, lengthening duration if rates are expected to fall, and shortening if rates are expected to rise.
b.A change in yield spreads across sectors would call for an intermarket spread swap, in which the manager buys bonds in the sector for which yields are expected to fall the most and sells bonds in the sector for which yields are expected to rise.
c.A belief that the yield spread on a particular instrument will change calls for a substitution swap in which that security is sold if its yield is expected to rise or is bought if its yield is expected to fall relative to the yield of other similar bonds.
24. While it is true that short-term rates are more volatile than long-term rates, the longer duration of the longer-term bonds makes their rates of return and prices more volatile. The higher duration magnifies the sensitivity to interest-rate savings.
25.The minimum terminal value that the manager is willing to accept is determined by the requirement for a 3% annual return on the initial investment. Therefore, the floor equals $1 million (1.03)5 = $1.16 million. Three years after the initial investment, only two years remain until the horizon date, and the interest rate has risen to 8%. Therefore, at this time, the manager needs a portfolio worth $1.16 million/(1.08)2 = $.9945 million to be assured that the target value can be attained. This is the trigger point.
26.The maturity of the 30-year bond will fall to 25 years, and its yield is forecast to be 8%. Therefore, the price forecast for the bond is $893.25 [n = 25; i = 8; FV = 1000; PMT = 70]. At a 6% interest rate, the five coupon payments will accumulate to $394.60 after 5 years. Therefore, total proceeds will be $394.60 + $893.25 = $1,287.85. The 5-year return is therefore 1,287.85/867.42 = 1.4847. This is a 48.47% 5-year return, or 8.23% annually.